Spatially varying heat flux driven close-contact melting – A Lagrangian approach

Abstract

Close-contact melting refers to the process of a heat source melting its way into a phase-change material. Of special interest is the close-contact melting velocity, or more specifically the relative velocity between the heat source and the phase-change material. In this work, we present a novel numerical approach to simulate quasi-steady, heat flux driven close-contact melting. It extends existing approaches found in the literature, and, for the first time, allows to study the impact of a spatially varying heat flux distribution. We will start by deriving the governing equations in a Lagrangian reference frame fixed to the heat source. Exploiting the narrowness of the melt film enables us to reduce the momentum balance to the Reynolds equation, which is coupled to the energy balance via the velocity field. We particularize our derivation for two simple, yet technically relevant geometries, namely a 3d circular disc and a 2d planar heat source. An iterative solution procedure for the coupled system is described in detail and discussed on the basis of a convergence study. Furthermore, we present an extension to allow for rotational melting modes. Various test cases demonstrate the proficiency of our method. In particular, we will utilize the method to assess the efficiency of the close-contact melting process and to quantify the model error introduced if convective losses are neglected. Finally, we will draw conclusions and present an outlook to future work.

Melting probe technology for subsurface exploration of extraterrestrial ice – Critical refreezing length and the role of gravity

“…An expression for the pressure is given by integrating the pressure gradient (17) subject to the pressure boundary condition (12), which yields…”

“…One potential loss is for example given by convective losses within the micro-scale melt film between the melting probe and the ice. The efficiency associated with these losses can either be studied experimentally [8] or through advanced modeling techniques [17] that go beyond the engineering design approach covered by (1).…”

Moment Approximations and Model Cascades for Shallow Flow

“…as evolution equation for the mean velocity u m , where we used the fact that the vertical coupling ω vanishes at the top and the bottom. By substituting for the boundary conditions ( 35) and (36) we weakly enforce the stick-slip condition…”

“…Since then similar mappings have been applied to a variety of processes, e.g. contact phase-change and ablation processes [25,36]. Comparing results of the reference system with the shallow flow moment models allows us for the first time to analyze and discuss the model error introduced during depth-averaging.…”

Modeling of hot-point drilling in ice

“…Some assumptions have been made regarding the behavior of the physical process in melt films: (1) the melting process is quasi-steady (Yoo and others, 1998; Groulx and Lacroix, 2003); (2) the flow in the melt film remains laminar (Groulx and Lacroix, 2003, 2007); (3) the melt film is very thin compared with the diameter of the thermal head, δ ≪ R , and a lubrication theory is valid for describing the melt film (Chen and others, 2008; Schüller and Kowalski, 2017); (4) viscous forces are dominant in the melt film, and ∂ 2/ ∂ h 2 ≪ ∂ 2/ ∂ s 2 (Chen and others, 1994); (5) the pressure in the melt film is uniform in the s -direction (Batchelor, 1967; Kumano and others, 2005b); (6) heat transfer in the melt film is dominated by conduction, and the heat transported through convective flow is negligible (Bahrami and Wang, 1987; Chen and others, 1994); (7) the temperature distribution in the melt film is linear (Groulx and Lacroix, 2003; Schüller and others, 2016); (8) the distribution of water velocity in the h -direction is paraboloidal; (9) the buoyancy force is ignored.…”